2 1 N ov 2 01 9 Canonical analysis of Holst action with first-class constraints only

We perform the canonical analysis of the Holst action for general relativity with a cosmological constant without introducing second-class constraints. Our approach consists in identifying the dynamical and nondynamical parts of the involved variables from the very outset. After eliminating the nondynamical variables associated with the connection, we obtain the description of phase space in terms of manifestly SO(3, 1) [or SO(4), depending on the signature] covariant canonical variables and first-class constraints only. We impose the time gauge on them and show that the Ashtekar-Barbero formulation of general relativity emerges. Later, we discuss a family of canonical transformations that allows us to construct new SO(3, 1) [or SO(4)] covariant canonical variables for the phase space of the theory and compare them with the ones already reported in the literature, pointing out the presence of a set of canonical variables not considered before. Finally, we resort to the time gauge again and find that the theory, when written in terms of the new canonical variables, either collapses to the SO(3) ADM formalism or to the Ashtekar-Barbero formalism with a rescaled Immirzi parameter.


I. INTRODUCTION
In the first-order formalism, (real) general relativity is described by the Holst action [1], which is made of the Palatini action coupled to the Holst term via the Immirzi parameter [2]. In vacuum (with or without a cosmological constant), this action reproduces exactly the same dynamics contained in the metric formulation of Einstein's theory obtained from the Einstein-Hilbert action as long as the orthonormal frame be nondegenerate. Outstandingly, the Holst action establishes the Lagrangian setting of the Ashtekar-Barbero variables [3], which constitute the building blocks of loop quantum gravity [4][5][6][7][8]. Nevertheless, the derivation of these variables makes use of the so-called time gauge, which breaks the Lorentz group SO (3,1) down to the SO(3) subgroup. This gauge fixing avoids the introduction of second-class constraints, simplifying the resulting canonical theory at the expense of local Lorentz invariance.
Because Lorentz invariance plays a fundamental role in modern physics, there have been different approaches tackling the Lorentz-covariant canonical analysis of the Holst action. Nonetheless, those perspectives introduce second-class constraints, which are dealt with at the end either by using the Dirac bracket [9] or by solving them explicitly [10][11][12][13]. Remarkably, in Refs. [11,12] the second-class constraints were solved while preserving the manifest Lorentz invariance of the theory, obtaining different sets of canonical variables for the phase space of general relativity that is now described by first-class constraints only.
In the standard approach, the second-class constraints are introduced due to a mismatch between the number of * merced@fis.cinvestav.mx † ljromero@fis.cinvestav.mx ‡ mcelada@fis.cinvestav.mx independent components of the Lorentz connection and those of the orthonormal field. It turns out that the spatial part of the connection corresponds to the configuration variables, and their canonically conjugate momenta are related to the spatial part of the orthonormal frame. Because the number of independent variables in the canonical momenta surpasses the number of components in the spatial part of the frame, one must add a quadratic constraint on the momenta in order to even things out. The Hamiltonian evolution of this constraint then generates a secondary constraint which, together with the former, makes up the set of second-class constraints of general relativity. Solving these constraints is what Refs. [10][11][12][13] are devoted to.
Alternatively, instead of real general relativity, one can move to the self-dual Palatini action [14][15][16] (obtained from the Holst action by taking the Immirzi parameter equal to the imaginary unit), which involves the self-dual part of the Lorentz connection. This action allows us to derive a Hamiltonian formulation of the theory (the Ashtekar formalism [17,18]) resorting neither to the introduction of second-class constraints nor to the time gauge [19] (see also [20]), thus preserving Lorentz invariance; however, as implied by the use of self-dual variables, the formulation is complex and needs to be supplemented with reality conditions [the variables take advantage of the isomorphism between the Lie algebras of SO(3, 1) and SO(3, C)].
So, in order to avoid complex variables and work with real ones, the introduction of second-class constraints to preserve Lorentz invariance seems inexorable. Is that so? In this paper we show that it is possible to perform the canonical analysis of the Holst action without introducing neither second-class constraints nor gauge fixings spoiling Lorentz invariance. This is accomplished by providing a parametrization of the spatial part of the connection that separates its dynamical components from its nondynamical ones. The former equal in quantity to the number of components of the spatial part of the frame (which in turn are related to the canonical momenta), whereas the latter appear quadratically in the action and then can be eliminated via their own equation of motion. The resulting canonical theory, being manifestly Lorentz covariant, agrees with the one reported in Ref. [11]; outstandingly, the derivation is much simpler than the original one and the geometrical meaning of the variables is clearer.
The outline of the paper is as follows. After establishing our conventions, we perform the 3+1 decomposition of the action in Sec. II, identifying the dynamical variables that make up the symplectic structure. We then reparametrize the spatial part of the connection in terms of these variables and some additional fields that turn out to be auxiliary fields. We get rid of the latter and arrive at the canonical formulation of general relativity with manifest Lorentz invariance and first-class constraints only. Later, we discuss the time gauge in Sec. III and the implementation of canonical transformations in Sec. IV. To close the paper, we give some conclusions.
Conventions. Spacetime indices are denoted by Greek letters (µ, ν, . . . ) so that points on the spacetime manifold M are labeled by coordinates {x µ } = {t, x a }, where t is the time coordinate (we use the "dot" notation for time derivatives when possible) and Latin letters at the beginning of the alphabet (a, b, . . . = 1, 2, 3) denote spatial indices. We assume that M has a topology R × Σ and foliate it by constant time hypersurfaces Σ t , each of which is diffeomorphic to some given orientable 3-manifold Σ without boundary. The coordinates {x a } label points on Σ t and from now on we just write Σ for any of these constant time hypersurfaces. Frame indices are associated with capital letters I, J, · · · = {0, i}, for i = 1, 2, 3. These indices are raised and lowered with the metric (η IJ ) = diag(σ, 1, 1, 1), where σ = −1 (σ = +1) in the Lorentzian (Euclidean) case. The frame rotation group corresponds to the Lorentz group SO(3, 1) for σ = −1 or to the rotation group SO(4) for σ = +1. The weight of a tensor is either indicated with the presence of a tilde over or below it, or mentioned somewhere else in the paper. The internal tensor ǫ IJKL and the spacetime tensor densityη µνλσ (η µνλσ ) are totally antisymmetric and such that ǫ 0123 = +1 andη t123 = +1 (η t123 = +1). In addition, we define the three-dimensional Levi-Civita symbols asη abc :=η tabc (η abc :=η tabc ) and ǫ ijk := ǫ 0ijk . The symmetrizer and antisymmetrizer are defined by V (αβ) := (V αβ + V βα )/2 and V [αβ] := (V αβ − V βα )/2, respectively. Furthermore, for an antisymmetric quantity V IJ we define its internal dual as * V IJ := (1/2)ǫ IJKL V KL and also the object where γ = 0 is the Immirzi parameter 1 . Its inverse is 1 We assume γ = ± √ σ, which means that the self-dual and anti-given by and satisfies (P −1 ) IJKL P KLMN = δ I [M δ J N ] . "∧" and "d" stand for the wedge product of differential forms and the exterior derivative, correspondingly.

II. CANONICAL ANALYSIS
In the first-order formalism, the orthonormal frame e I (assumed to be nondegenerate) and the SO(3, 1) [or SO(4)] connection ω I J are independent degrees of freedom that encode the gravitational field. In terms of them, the Holst action [1] for general relativity is given by where F I J := dω I J + ω I K ∧ ω K J is the curvature of ω I J , which is compatible with the metric η IJ , dη IJ − ω K I η KJ − ω K J η IK = 0 (and thus ω IJ = −ω JI ); κ is a constant related to Newton's constant and Λ is the cosmological constant. Adapted to the spacetime foliation, the frame and the connection can be written as e I = e t I dt + e a I dx a and ω I J = ω t I J dt + ω a I J dx a , respectively.
Let us introduce a vector n I with the following two properties at fixed t: e a I n I = 0 and n I n I = σ. Explicitly, this vector takes the form where q > 0 (of weight +2) is the determinant of the spatial metric q ab := e a I e bI , whose inverse metric is denoted by q ab . The projector on the orthogonal plane to n I is given by Geometrically speaking, since at each point of M the orthonormal frame e I can be thought of as an isomorphism between the coordinate basis and the orthonormal basis of tangent space, the vector n I corresponds to the normal vector to the hypersurface Σ with respect to the latter basis; likewise, tangent vectors to Σ are translated into orthogonal vectors to n I . Thus, the splitting of the tangent space into the orthogonal and parallel parts to n I encodes the spacetime foliation.
self-dual sectors are excluded in our approach.
The 3+1 decomposition of the action (3) yields (we recall that all spatial boundary terms will be neglected because Σ has no boundary) where F ab is the curvature of ω a I J , dtd 3 x is a shorthand for dt ∧ dx 1 ∧ dx 2 ∧ dx 3 , and we have defined and Notice that, in Eq. (6), both ω tIJ and e t I appear linearly in the action and thus play the role of Lagrange multipliers. It is customary to split the latter into the components parallel and orthogonal to n I . Thus, we can express it as where N is the lapse function and N a is the shift vector [21]. The term of the action (6) involving e t I then becomes the combination −N aṼ a −ÑZ for andÑ := N/ √ q. Let us introduce, for future purposes, the densitized metrich ab := q −1 q ab , whose inverseh ab is given byh ab = Π aIΠb I ; its determinant h := det(h ab ) has weight +4 and is related to q by h = q 2 . Equation (7) can be inverted to express e a I in terms ofΠ aI , yielding which allows us to express the vector n I as In addition, we also define the connection ∇ a compatible with e a I that satisfies These are 36 equations for 18 unknowns Γ c ab (= Γ c ba ) and 18 unknowns Γ aIJ (= −Γ aJI ). Their solution is the Christoffel symbol Γ c ab for the spatial metric q ab and with n I given by (4). Notice that Eqs. (7) and (13) imply that ∇ a annihilatesΠ aI too: In terms ofΠ aI , the expression (14) becomes with n I given by (12). The curvature of Γ a I J is R ab Note that the first term on the right-hand side of the action (6) involves the time derivative of (γ) ω aIJ and thus contributes to the symplectic potential of the theory.
Since there are 18 independent components in (γ) ω aIJ (the same as in ω aIJ ), the usual approach requires to introduce the same number of canonically conjugate momenta. However, since these momenta are built up from the 12 components e a I , six additional constraints on the momenta must be added [10][11][12][13]. This is the traditional path taken and it leads to the emergence of second-class constraints; one set being the aforementioned constraints on the momenta and the other arising from the preservation under time evolution of the former. Here, we will follow a different path inspired by our previous work [11]; instead of introducing constraints on the momenta, we will properly parametrize the 18 components of ω aIJ into 12 dynamical and six nondynamical variables. The 12 dynamical variables thus will correspond to the configuration variables, whereas the six nondynamical ones will be later eliminated from the action, obtaining at the end a manifestly SO(3, 1) [or SO(4)] covariant canonical formulation of the Holst action. To do this, note that the symplectic term in Eq. (6) can be expressed as 2 where we have introduced the 12 configuration variables C aI defined by which can be thought of as an operator that singles out the 12 configuration variables C aI -constructed out of the components of the connection ω aIJ -that contribute to the symplectic structure. The expression (18) can be solved for (γ) ω aIJ to express the connection in terms of C aI plus six additional variablesλ ab (=λ ba ) living in the kernel of W a b IJK : In addition, we introduce the tensor densityŨ ab cIJ (= U ba cIJ = −Ũ ab cJI ) defined as where the internal dual in Eq. (22) acts on either the first pair or the last pair of indices of P −1 ( * and P −1 commute with one another). Together, the objects (19), (21a), (21b) and (22) satisfy the following orthogonality relations: as well as the completeness relation Therefore, W andŨ are orthogonal projectors that allow us to split the 18 components of the connection ω aIJ (or (γ) ω aIJ ) into the 12+6 variables (C aI ,λ ab ). The associated decomposition given in Eq. (20) is induced by the symplectic potential (17). The inverse of the map (20) is given by Eq. (18) together with which clearly shows that C aI andλ ab are independent variables among themselves. Substituting Eq. (20) into Eqs. (8), (10a) and (10b), we obtain, after some algebra, where the coefficients G abcd :=h abhcd −h (a|ch|b)d have weight +4. Notice that there no terms involvingλ ab in Eq. (26a), and that Eqs. (26b) and (26c) depend onλ ab but not in their derivatives. This is staggering, since the original expressions (10a) and (10b) involve derivatives of the connection. The action (6) now takes the suggestive form which really resembles what is expected when casting an action in Hamiltonian form. Nevertheless, we have not finished yet because the action (27) still depends on the variablesλ ab as indicated above. However, the variables λ ab are auxiliary fields [22] that can be eliminated by setting the variational derivative of the action (27) with respect toλ ab equal to zero: This equation is linear inλ ab and can be solved for them as long asÑ = 0, which is always fulfilled since the orthonormal frame was assumed to be nondegenerate. Hence, the solution forλ ab is Substituting (29) back into the action (27), integrating by parts the term involving the covariant derivative in Eq. (26c), and collecting all the terms proportional toG IJ , the action acquires the final form whereG IJ is the same as in Eq. (26a), whereas and Since λ IJ (or ω tIJ ), N a andÑ appear linearly in the action (30), they play the role of Lagrange multipliers and imposeG IJ ,D a andH as constraints, respectively. These constraints, known correspondingly as the Gauss, diffeomorphism and scalar constraints, are the same as the ones found in Ref. [11] by solving the second-class constraints of general relativity in a manifestly SO (3,1) [or SO(4)] covariant fashion. Therefore, by following a different path along which the introduction of secondclass constraints in the theory is utterly avoided, we have arrived at the same Hamiltonian formulation of general relativity. It is worth mentioning that in Ref. [11] there is a sign ambiguity ǫ in the solution of the second-class constraints (since they are quadratic in the canonical momenta) that later propagates along the canonical analysis; such an ambiguity was completely avoided in the present work because no second-class constraints were introduced here.
To sum it up, from the initial 16 free variables contained in the orthonormal frame e µ I , four of them, associated to the time component e t I , play the role of Lagrange multipliers that impose the diffeomorphism and scalar constraints,D a ≈ 0 andH ≈ 0 respectively, whereas the remaining 12 components e a I of the frame are absorbed into the canonical variableΠ aI , which is related to them by Eq. (7) or Eq. (11). On the other hand, from the initial 24 variables in the connection ω µIJ , the six components ω tIJ are involved in the Lagrange multipliers λ IJ that impose the Gauss constraintG IJ ≈ 0, the six variablesλ ab are auxiliary fields fixed by their own equation of motion and given by Eq. (29), and the remaining 12 variables C aI constitute the configuration variables that, together withΠ aI , make up the canonical variables of the theory; according to Eq. (30), they are normalized such that the fundamental Poisson bracket reads {C aI (t, x),Π bJ (t, y)} = (1/2κ)δ b a δ J I δ 3 (x, y), where δ 3 (x, y) is the three-dimensional Dirac delta.
Also, notice that in the case of a vanishing cosmological constant (Λ = 0), the formulation described by the action (30) is invariant, up to a global factor, under a constant rescaling of the momenta variablesΠ aI → ΩΠ aI , with Ω being a nonvanishing real number, since both the internal vector n I and the connection Γ aIJ are left invariant by this change [see Eqs. (12) and (16), respectively]. Thus, rescalingΠ aI together with a redefinition of the Lagrange multipliers λ IJ → Ω −1 λ IJ , N a → Ω −1 N a , andÑ → Ω −2Ñ , leaves the action (30) almost unaltered, because the constraints (26a), (31a), and (31b) remain the same; however, the theory now obeys the fundamental Poisson bracket {C aI (t, x),Π bJ (t, y)} = (1/2κΩ)δ b a δ J I δ 3 (x, y). This property has already been exploited within the time gauge framework [7]. Here, we just showed that it is a distinctive feature of the Hamiltonian formulation of general relativity without a cosmological constant, regardless of any gauge fixation.
Furthermore, we also point out that it is not necessary to split e t I into lapse and shift components, which provides a way of unifying the vector and scalar constraints into one SO(3, 1) [or SO(4)] covariant constraint that, up to terms proportional to the Gauss constraint, takes the formH Hence,H I andG IJ constitute the only constraints of the theory, and whereas the latter generates local SO(3, 1) [or SO(4)] transformations, the former is related to spacetime diffeomorphisms.

III. TIME GAUGE
The time gauge fixes the freedom to perform boost transformations and leaves a remnant SO(3) gauge symmetry. The time gauge is imposed by hand through the constraintΠ a0 ≈ 0, which weakly commutes-in Dirac's sense [23]-with all the constraints except withG i0 ≈ 0 (boost generator), for which the Poisson bracket gives This renders the pair (Π a0 ,G i0 ) second class becauseΠ ai is an invertible 3 × 3 matrix that is associated with the densitized triad through Eq. (7). We make te secondclass constraints strongly equal to zero. From Eq. (26a), the solution ofG i0 = 0 is whereΠ ai is the inverse ofΠ ai . Likewise, the time gauge implies n 0 = sgn(det(Π ai )) and n i = 0 from Eq. (12), and Γ a0i = 0 from Eq. (16). Moreover, the SO (3) indices are raised and lowered with the Euclidean metric δ ij . Let us define the SO(3) connection Γ ai := −(1/2)ǫ ijk Γ a jk ; equation (15) then implies that Γ ai is the connection compatible withΠ ai : ∇ aΠ bi = ∂ aΠ bi + Γ b acΠ ci −Γ c acΠ bi +ǫ ijk Γ ajΠ b k = 0. Its curvature is given by and it describes the intrinsic geometry of Σ. According to Eq. (16), Γ ai is given explicitly by Therefore, in the time gauge, the action (30) reduces to where λ i := −(1/2)ǫ ijk λ jk . The SO(3) Gauss constraint G i := −(1/2)ǫ ijkG jk and the diffeomorphism and scalar constraints take the form From Eq. (38a) we infer that the object A ai := −n 0 γC ai is an SO(3) connection and we can define its field strength by In terms of the connection A ai , the action (37) reads where we have used the identity ) to rewrite the last term of Eq. (38c). Integrating by parts the last term inH and redefining the Lagrange multiplier in front of the Gauss constraint as µ i := λ i + (n 0 /γ)Π a i ∇ aÑ , we get withC Thereby, we have straightforwardly arrived at the Ashtekar-Barbero formulation for general relativity with cosmological constant [1,3] (see also Ref. [7]). From Eq. (42), we can read off the Poisson bracket {A ai (t, x),Π bj (t, y)} = (−n 0 γ/2κ)δ b a δ j i δ 3 (x, y). Notice that, in order to coincide with the results of Ref. [1], we must take n 0 = −1 in the Lorentzian case; this amounts to taking det(Π ai ) > 0, as in the analysis carried out by Holst.
Alternatively, by using again Eq. (41) we can get rid of the term involving R ab i in the scalar constraint (43), to getC with Substituting (44) into the action (42) and integrating by parts the last term in (44), we get where ρ i := µ i + n 0 (σγ − γ −1 )Π a i ∇ aÑ . This alternative form of the scalar constraint agrees with the one reported in Ref. [7]. As usual, instead of the diffeomorphism constraintD a , we can use the vector constraint As shown in our previous work [11], the manifestly SO(3, 1) [or SO(4)] covariant formulation of the Holst action contained in Eq. (30) can, alternatively, be expressed in terms of other manifestly SO(3, 1) [or SO(4)] covariant canonical variables. The underlying canonical transformations are such that they leave the canonical momentã Π aI unchanged, whereas the configuration variables are promoted to new ones. They can be encompassed by the map (C aI ,Π aI ) → (X aI ,Π aI ), where X aI is given by where α and β are continuous real parameters. The transformation is indeed canonical, since the symplectic potential in Eq. (30) changes by a boundary term: In the new variables, the action (30) acquires the form where the Gauss, diffeomorphism and scalar constraints are given bỹ respectively. Notice that the diffeomorphism constraint takes exactly the same form as the original one and that it is independent of both α and β; this means that X aI transforms as a 1-form under spatial diffeomorphisms for any choice of α and β. On the other hand, the Gauss and scalar constraints strongly depend on the values of these parameters. However, if β = 1 the Immirzi parameter γ drops out from Eqs. (51a) and (51c) regardless of the value of α. We conclude by analyzing some particular nontrivial cases of the canonical transformation (48) (α = 0 = β is the identity transformation) i) For α = 1 = β the configuration variable X aI becomes the configuration variable Q aI introduced in Ref. [11].
The Gauss and scalar constraints simplify considerably in terms of the phase-space variables (Q aI ,Π aI ). Notice that the Immirzi parameter does not appear in the constraints, which take exactly the same form as those arising in the manifestly SO(3, 1) [or SO(4)] covariant canonical analysis of the Palatini action [24].
ii) For α = 1 and β = 0 the configuration variable X aI becomes the configuration variable K aI introduced in Ref. [11]. The Gauss and scalar constraints simplify a bit in terms of the phase-space variables (K aI ,Π aI ), but the Immirzi parameter is still present.
iii) For α = 0 and β = 1 the configuration variable X aI will be denoted as q aI . This case had not been previously reported in literature and leads to new phase-space variables (q aI ,Π aI ). In terms of them, the constraints (51a)-(51c) take the form Notice that these constraints are independent of the Immirzi parameter too. Hence, the configuration variables q aI -together with the configuration variables Q aI -are naturally associated with the Palatini action [25]. Time gauge. In the time gauge, the canonical transformation (48) becomes, after using Eq. (35), the latter being independent of α (this parameter only appears in the expression found for X a0 , whose value is not relevant in the time gauge). Moreover, from Eq. (51a) the SO(3) Gauss constraint,G i = −(1/2)ǫ ijkG jk , be-comesG whereas the constraints (51b) and (51c) read respectively. Therefore, the action (50) takes the form Notice that the action is independent of α, whereas the value of β determines the canonical theory under consideration, according to whether β = 1 or β = 1. Let us analyze this in detail: a) For β = 1, the constraints (54), (55a) and (55b) becomeG which can be recognized as the constraints of the SO(3) ADM formalism [19]. Note also that Eq. (53b) is the converse of Barbero's transformation since we go from the connection A ai to the vector X ai . This means that, for β = 1 and regardless of the value of α, we obtain the SO(3) ADM formulation of general relativity, where −n 0 X ai is an object closely related to the extrinsic curvature. As already mentioned, the configuration variables q aI (α = 0) and Q aI (α = 1) belong to the case β = 1. Therefore, from (53a) and (53b), they are set to q a0 = −σn 0Πbi ∂ bΠai , Q a0 = 0 and q ai = −(n 0 /γ)(A ai − Γ ai ) = Q ai in the time gauge. b) For β = 1, note that Eqs. (54), (55a) and (55b) resemble Eqs. (38a)-(38c) with γ/(1 − β) taking the place of γ. In fact, we can express the Gauss constraint (54) asG where we have identified the SO(3) connection A ai := −n 0 γX ai /(1 − β). Let us denote its field strength by Then, using the identity we rewrite the last term of Eq. (55b), so the action (56) becomes with Integrating by parts the last term of Eq. (61b), the action (60) finally acquires the form where ν i := λ i + [n 0 (1 − β)/γ]Π a i ∇ aÑ and Notice that the constraints (58), (61a) and (63) take exactly the same form as (40a), (40b) and (43), respectively, with A ai and γ replaced correspondingly with A ai and γ/(1 − β). Therefore, for β = 1 we obtain the Ashtekar-Barbero formulation with a rescaled Immirzi parameter γ/(1 − β). It is worth realizing that the connections A ai and A ai are related to each other by Note that they are the same for β = 0, as expected. As already mentioned, the configurational variable K aI (α = 1 and β = 0) belongs to the case β = 1. Therefore, from (53a) and (53b), it is set to K a0 = 0 and K ai = −n 0 γ −1 A ai in the time gauge.

V. CONCLUSIONS
In this paper, we have carried out from scratch the canonical analysis of the Holst action without introducing second-class constraints. Our strategy consisted in splitting the 18 degrees of freedom contained in the spatial connection ω aIJ , by means of the orthogonal projectors (19) and (22), into 12 dynamical SO(3, 1) [or SO(4)] covariant variables C aI plus six additional fields λ ab , something explicitly exhibited in Eq. (20). When this expression is substituted into the action, we realize that the latter are auxiliary fields that can be easily eliminated using their own equation of motion. Doing so, the action becomes Eq. (30), in which the canonical pair (C aI ,Π aI ) is subject to the first-class constraints (26a), (31a) and (31b), namely the Gauss, diffeomorphism and scalar constraints, respectively. It is really remarkable how our approach simplifies the canonical analysis of general relativity, not only leading to a manifestly SO (3,1) [or SO(4)] covariant parametrization of the phase space of the theory (which agrees with the one found by solving second-class constraints [11]), but also allowing us to keep track of the role played by each one of the original variables involved in the Holst action. Afterwards, we imposed the time gauge and showed that it immediately leads to the Ashtekar-Barbero variables. In that process, the spatial components −n 0 γC ai get identified with the Ashtekar-Barbero connection.
In addition, we discussed the set of canonical transformations given by Eq. (48) which depends on two parameters α and β, and relates different SO(3, 1) [or SO (4)] covariant parametrizations of the phase space of general relativity; some particular cases of them-(Q aI ,Π aI ) and (K aI ,Π aI )-have already been reported in the literature [11]. We also reported the new canonical phasespace variables (q aI ,Π aI ), for which the constraints are given by (52a)-(52c) and turn out to be independent of the Immirzi parameter. In the time gauge, the canonical theories associated to these variables bifurcate into two kinds depending on the value of β (α is not important in the time gauge): for β = 1 we obtain the SO(3) ADM formulation of general relativity, whereas for β = 1 we arrive at the Ashtekar-Barbero formulation with a rescaled Immirzi parameter γ/(1 − β).
Although in this paper we have discussed the canonical analysis of the Holst action with respect to a foliation by spacelike hypersurfaces, our strategy can also be adapted to deal with the case of a timelike foliation as well for Lorentzian signature. In that case, the vector n I is normalized such that n I n I = 1 and we can follow a procedure very similar to that described in Sec. II. After eliminating the auxiliary fields, we get the same Lorentzcovariant variables and first-class constraints than those obtained in Ref. [12] after solving the second-class constraints. Then, by imposing the space gauge there, the Ashtekar-Barbero formulation with gauge group SU (1, 1) as well as the SO(2, 1) ADM formulation arise. The detailed analysis of all of this will be reported in another paper.
Our procedure stands out for its simplicity, methodology, and economy. Whereas in Dirac's approach the canonical analysis can be cumbersome due to the amount of variables involved in the formalism, here we have identified from the very beginning the fundamental variables and gotten rid of the superfluous ones in the process. We point out that the same theoretical framework can also be applied to the n-dimensional Palatini action [25] and also to the case of the coupling of matter fields to the Holst action. These results will be reported elsewhere too.
Regarding the unified constraint (33) at the end of Sec. II, it would be really interesting to compute the gauge algebra satisfied by this constraint together with the Gauss constraintG IJ and also to establish the relation between the gauge transformations generated byH I and the alternative gauge transformations for the Holst action reported in Ref. [26].
We think our results can also be useful for investigating the asymptotic behavior of the gravitational field where Lorentz invariance is relevant as well as for the study of gravitational field configurations using numerical methods, where our approach could provide new insights for positing initial value problems in gravity because it preserves Lorentz invariance.
Finally, although the canonical variables contained in this paper are no longer connection variables-but they give rise to connection variables in the time gauge, as pointed out above-and thus their implementation in the quantum theory is quite nontrivial, our results should motivate the development of new mathematical techniques to potentially use these variables in applications to quantum gravity.